In-depth investigations of size and occupancies in cobalt ferrite nanoparticles by joint Rietveld refinements of X-ray and neutron powder diffraction data

Combined neutron and X-ray powder diffraction investigations of CoFe2O4 are reported, aimed at investigating the robustness, reproducibility and reliability of structural parameters from Rietveld refinement.


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Lorentzian isotropic size parameter, Y, was constrained between patterns taking into account the differences 32 in broadening as function of scattering angle due to the use of different wavelengths. This is illustrated in Table  33 S1, where pattern #1 refers to Cu Kα1, pattern #2 to Co Kα1 and the last by DMC neutron source. The 34 wavelength of the first pattern was chosen as default. The wavelength ratio between a pattern (#2 or #3) and 35 the default pattern (#1) was used to describe the code to apply in the FullProf Suite software. This leads to 36 λCo/λCu= 1.161 and λDMC/λCu= 1.596 for Co and DMC patterns, respectively. Furthermore, the value for the 37 Lorentzian isotropic size parameter for the two last patterns has to be calculated using the wavelength ratio 38 too. Hence, equations (S 1) and (S 2) leads to Y value of the two last patterns. 39 = * λ λ = 0.424557 * 1.161 = .
(S 2)   The ACS is used in the FullProf Suite software, 2,3 and is described as: 52 Where < > is the volume-weighted average domain size in the direction of the scattering vector, λ is the 53 wavelength of the X-ray source, is the integral breadth of the H th reflection and the Bragg angle of 54 the H th reflection. 55 56

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To determine the net intrinsic magnetisation (Msat Neutron ), the atomic fraction of each atoms in the inverse spinel 58 structure, (Co 2+ 1-x Fe 3+ x) tet [Co 2+ x Fe 3+ 2-x] oct O4, and the refined magnetic moment dipole moment Rx of both 59 tetrahedral (Td) and octahedral (Oh) sites were considered as a function of the the formula unit (f.u.) CoFe2O4. 60 We used the following equations: 61 with . . the formula unit magnetic moment, Mf.u. the molecular mass of the formal unit (234.625 g/mol), NA 62 the Avogadro constant with NA = 6.022.10 23 mol -1 and µB = 0.927.10 -23 A.m². 63 64 Comparative study on the effect of the Co:Fe magnetic moment ratio 65 When refining the magnetic structure of CoFe2O4 an important question need to be asked: should we take into 66 account the Co orbital contribution (μS+L(Co)) to the magnetic moment of Co and Fe? To answer that question 67 we have investigated the effect of the Co:Fe magnetic moment ratio. Two extrem cases were considered: 68 1) the cobalt orbital moment is supposed to be quenched, therefore only the number of unpaired electrons is 69 used to described the magnetic moment of Co and Fe, which is 3 and 5, respectively. 2) The orbital contribution 70 of cobalt is considered, leading to Fe 3+ having a magnetic moment of 5.9 (μS(Fe)), and Co 2+ of 5.2 (μS+L(Co)). 4 71 A third refinement model was also investigated, where the constrainement of Td and Oh was raised, allowing 72 both sites to be refined independently. The data used for this study were those from AC240 sample. The result 73 of this investigation is summarized in Table S3 below. 74

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The Rx ratio modification does not interfere with the refinement of the structural properties, since the unit cell, 80 oxygen position, ADP and occupancies are exactly the same as the previous study above. 81 Undoubtedly, changing the initial 3:5 (0.6) ratio to 5.2:5.9 (0.88) increase the atomic magnetic moment of Co, 82 but for Fe it was lowered, compared to AC240. Therefore, m and M Neutron are increased for AC240_Rx(5.2:5.9), initial model, while Rx(Co 2+ ) has increased. Nonetheless, Rx(Fe 3+ ) Td is within the same uncertainty as the first 86 model, which is not the case for iron in Oh sites. Therefore, the calculated m and M Neutron were both been 87 reduced by ~20% compared to AC240. Thus, refining both sites individually does lead to a proper description 88 of the magnetic phase of CFO, since the calculated magnetic moment M Neutron does not correspond to the 89 measured value Msat VSM . 90 In the end, two extreme cases were tested to describe the magnetic phase of CoFe2O4: a model based on the 91 number of unpaired electron of both cobalt and iron, and another where the orbital magnetic moment is 92 included. Regarding the refinement, it cannot be conclude which model is better. However, by comparing with 93 our experimental data, we could argue that our initial model is in good agreement with macroscopic 94 magnetisation data. Thus, our approximation of the orbit moment being quenched and the value of the magnetic 95 moments being proportional to the number of unpair electrons independent of temperature is an adequate 96 approximation. 97

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In the present work, the PXRD patterns collected with the Co and Cu sources contain the same 18 reflections 100 while the neutron data from DMC is limited to only 6 reflections. Three different weighting schemes have 101 been evaluated for the combined refinement of all three patterns; First, an 'equal weight (Ew)' scheme was 102 applied with the three patterns: DMC/Co/Cu being weighted 0.33/0.33/0.33. Secondly, an 'information 103 weighted (Iw)' scheme was employed based on the number of reflections in each pattern divided by the sum 104 of all reflections Iwpattern = Σpeaks in one pattern/Σpeaks in all patterns, yielding a 0.14/0.43/0.43 weighting of the 105 DMC/Co/Cu datasets. Finally, an 'arbitrary weight (Aw)' scheme was chosen to favour the weight of the 106 neutron data over the two PXRD sources, with a weighting equal to 0.5/0.25/0.25. The powder diffraction 107 patterns refined using the Ew model are shown in the manuscript ( Fig. 1), while Figure S1 display the two 108 other wheighting schemes. Notably, it is not possible to visually distinguish the refinement models based on 109 the three weighting schemes. Iw -0.14 / 0.43 / 0.43 Q (Å

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The three models give very similar results and/or are within the uncertainty of each other, indicating that the 118 weight scheme does not hugely impact the refinements of the present data. It is observed that increasing the 119 weight of the two PXRD sources slightly increases the obtained unit cell length, crystallite size and oxygen 120 position, while the occupancy of Co 2+ is lowered in the Td sites. The inversion degree is also higher with an 121 increase of the PXRD data weight. The thermal vibrations, as well as the three magnetic parameters (Rx, m and 122 M), are not affected by the weight modification as only the neutron data is providing information about these 123 parameters. 124 Table S4: Comparison of three weighting schemes and their impact on the refined structural and magnetic parameters.  source not found., we clearly see that the R-factors of the Cu pattern are not influenced by the weight 136 modification, contrary to Co and DMC patterns. However, the changes are small and does not exceed 1%. As 137 shown here, simply using the agreement factors may not be an appropriate way to determine the optimal 138 weighting scheme. In some cases, the obtained fit might get worse because the strengths of the included or 139 more heavily weighted dataset highlights the shortcomings of the employed model. In that case, blindly trusting 140 the R-factors will yield the best fit, but not necessarily an accurate result. Instead, it may be more intuitive to 141 weight the patterns according to their individual strengths and the desired structural information. The Cu 142 pattern carries the least information about the Co/Fe occupancies in the spinel structure. However, the Cu and 143 Co patterns are rather important for the description of the lattice and microstructural parameters since they 144 have the better peak and Q-range resolution. Consequently, to investigate the spinel inversion degree, as well 145 as the structural properties, it is preferable to weight the neutron data higher in the refinement, but without 146 neglected the PXRD data. 147

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The diffraction pattern of the six remaining combination are display in Figure S2   The PXRD and NPD data of samples A, B and C are plotted in Figures S3-S5, respectively. The additional 158 peak observed around 3.1 Å -1 for C_3 was attributed to the (111) reflection of pure Ni, coming from a 159 thermocouple used during the diffraction experiment. The peak present at 5.4 Å -1 in all PXRD data is attributed 160 to the (222) reflection of the Al sample holder and was exclude from the refinement to improve the fit. The 161 down sloping background starting at about Q = 5 Å -1 suggests a reduction in the probed sample volume, due 162 to the beam fully penetrating the sample and hitting the aluminium sample holder below. This will cause the 163 ADPs to be slightly overestimated. 164 In order to investigate the effect of the thermal vibration on the refinement of the atomic and microstructural 165 parameters of CoFe2O4, the thermal vibration of sample C was fixed at 1.57 Å², corresponding to the refined 166 value obtained in sample A. This constitute what we have named C_BovFIX sample. The results of the 167 refinement of the data from sample C and sample C_BovFIX are gathered in Table S5. 168 The Bov investigation has revealed that the thermal vibrations did not modify the refinement of C_BovFIX 169 sample. Only minute deviations were recorded but considering the uncertainties, the deviation is too small to 170 confirm that changing Bov impacts the refinement.  Figure S3: a), b) and c) are the PXRD data of samples A1, A2 and A3, respectively, while d), e) and f) represent the NPD 177 data. The experimental data is shown by the red dots, the refined model by the black line and the residual by the blue line.

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The weighted profile and Bragg R-factors are indicated for each diffraction pattern. Frequencies of 3 and 15 data points 179 were selected to plot the NPD and PXRD patterns, respectively.    Site occupancy: All samples have the same tetrahedral site occupancy, with Fe 3+ occupying 70%, which is 214 close to a random occupancy (x = 2/3) for a stoichiometry of Fe:Co = 2:1. 10,11 Different refined occupancies 215 vacancy structure would also be expected to have a smaller unit cell. 12 Notably, spinel ferrite nanoparticles 219 have previously been reported to have vacancies within the Oh site. As the Oh site is fixed to be fully occupied, 220 the refinement can only reduce the site scattering power by introducing more Co 2+ on the Oh site, or increase 221 the ADP. Vacancies would also reduce the magnetic moment. 222 Since the ADPs are linked to the occupancy, it is therefore logical to see variation in the magnetic properties. 223 By using two Biso, the occupancy of Td sites have a trend to be equal to a 30:70 ratio, while for Oh sites the 224 occupancy is between 30:70 and 40:60, depending on the sample. Undoubtly, the fraction of Fe in the 225 refinement is increased by using the second model.

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The powder diffraction patterns for the different Co-precursors are shown in Figure S6. 248